I have a brother who became a Christian at a young age and when he was a teenager he turned rebellious. He is still rebelling over a decade later. He does not profess to be a Christian, he does not attend any church body or even desire to be a part of the Christian faith. How do I treat him -- as an unbeliever or as a believer in I Corinthians 5:9-11. Would you explain this passage too?
Your brother knows what Christ teaches and chose to reject it. He is a former believer who fell away. Such a person is harder to bring back to the truth because there isn't anything new that you can offer him that he doesn't already know. "For if, after they have escaped the pollutions of the world through the knowledge of the Lord and Savior Jesus Christ, they are again entangled in them and overcome, the latter end is worse for them than the beginning. For it would have been better for them not to have known the way of righteousness, than having known it, to turn from the holy commandment delivered to them" (II Peter 2:20-21). Not that you shouldn't try, but realize that there isn't anything you can do to force him to change. He has to decide to return on his own.
The point Paul is making in "I wrote to you in my epistle not to keep company with sexually immoral people. Yet I certainly did not mean with the sexually immoral people of this world, or with the covetous, or extortioners, or idolaters, since then you would need to go out of the world. But now I have written to you not to keep company with anyone named a brother, who is sexually immoral, or covetous, or an idolater, or a reviler, or a drunkard, or an extortioner--not even to eat with such a person" (I Corinthians 5:9-11) is that as Christians we cannot give approval to sin, even indirectly. The world is filled with sin, yet we must live in it without letting that sin impact our lives.
But a brother who is in sin is more dangerous because typically the barriers are down between brethren and sin can spread more easily. "Your glorying is not good. Do you not know that a little leaven leavens the whole lump?" (I Corinthians 5:9). In Corinth was a man who was involved in sexual sins that even Gentiles thought were horrible. Yet, the Corinthians only showed this sinning brother acceptance. They compromised their standards and thus compromised themselves.
Nor was their treatment of this fallen brother an encouragement for him to change. Why should he leave his sin when he has all the benefits of acceptance while remaining in sin?
You treat your brother politely. When opportunities come to encourage him to come back to the Lord, use them, but never in your actions, directly or indirectly, give the impression that you accept his foolish behavior. He left you and what you love, that doesn't mean you must chase after him. He needs to see that he's lost a lot of benefits and needs to return to the Lord.